Math Insight

Double integrals as area

Suggested background

One use of the single variable integral is calculate
the area under a curve $f(x)$ over some interval $[a,b]$
by integrating $f(x)$ over that interval.

Sometimes, we use double integrals to calculate area as well.
But, the approach is quite different.
It's fairly simple to see the trick to accomplish this once you
can imagine how to use a single integral to calculate
the length of the interval.

What happens if you integrate the function
$f(x)=1$ over the interval $[a,b]$? You can calculate that
\begin{align*}
\int_a^b f(x)dx = \int_a^b 1 dx = x\big|_a^b = b-a.
\end{align*}
The integral of the function $f(x)=1$ is just the length of the
interval $[a,b]$. It also happens to be the area of the rectangle of
height 1 and length $(b-a)$, but we can interpret it as the length of
the interval $[a,b]$.

We can do the same trick for double integrals.
The integral of a function $f(x,y)$ over a region $\dlr$ can be
interpreted as the volume
under the surface $z=f(x,y)$ over the region $\dlr$.
As we did above, we can try the trick of integrating the
function $f(x,y)=1$ over the region $\dlr$.
This would give the volume under the function $f(x,y)=1$ over $\dlr$. But
the integral of $f(x,y)=1$ is also the area of the region $\dlr$. This
can be a nifty way of calculating the area of the region $\dlr$. If we
let $A$ be the area of the region $\dlr$, we can write this as
\begin{align*}
A = \iint_\dlr dA.
\end{align*}

Example

Find area of region bounded by parabola $x=y^2$ and the
line $y=x$. The region is pictured below.

Solution: We'll let $y$ go from 0 to 1. Then $x$ goes from $y^2$ to $y$.